Created by Prof. Dr. Thomas Hotz, Stefan Heyder, Matthias Glock and Sebastian Semper of the AG Stochastik, Technische Universität Ilmenau
in collaboration with Prof. Dr. Alexander Krämer and Anne Böhle of the School of Public Health, Bielefeld University.
Above, we show results for the all countries with at least 30 cases 7 days ago; results for the world as well as the largest 20 countries may be found here. In addition, results for Germany and its Bundesländer are available here (in German).
We estimate the (effective) reproduction number R(t) at day t, i.e. the average number of people someone infected at time t would infect if conditions remained the same.
The estimator has been taken from (Fraser 2007). It compares the number of infections at a time point with the number of infectious cases at that time, weighted by their respective infectivity. Note that constant (per country) underreporting does not affect the estimator since both the number of infections and the number of infectious are reduced by the same proportionality factor.
For this estimator, we derived (approximate, pointwise) 95% confidence intervals using the delta method.
However, the size of the confidence intervals reflects only those statistical uncertainties due to the random dynamics of the epidemic. But since the estimator is based on assumptions about the infectivity of the virus, and given that the data are not perfect because of a change of reporting criteria, varying amounts of testing etc., the estimates should be interpreted cautiously and not be taken at face value. Still, we believe that one can draw qualitatively credible conclusions from them.
Estimates are shown in black, confidence intervals as grey stripes, with values specified by the left axis (on a log-scale).
The critical value for the reproduction number is 1, shown as a red horizontal line: a value larger than one would result in an exponential increase of infections, a value smaller than one in a decrease.
The analysis is based on newly reported cases of Coronavirus Disease 2019 (COVID-19) per day, shown as blue bars as specified by the right axis (on a linear scale). For these we rely on the data provided by Johns Hopkins University.
For the estimated reproduction number (lines, left vertical axis), the horizontal axis specifies the corresponding date of infection whereas for the newly reported cases (bars, right axis), it specifies the date the cases were reported. Mondays are labelled.
The graphics are updated daily (last update: 04/04/20 15:31 GMT), showing data up to yesterday.
Note that cases are reported much later than the corresponding day of infection, namely after incubation time plus some more days necessary for testing and reporting the case to the authorities. For simplicity we assume that cases are reported 7 days after infection. Therefore, estimates for the reproduction number lag one week behind the reporting of new cases.
In a population where no countermeasures have been put into place, the so-called basic reproduction number R0 is believed to be given by some value between 2.4 and 4.1 (Read et al. 2020). Estimates higher than that might be explained by a considerable number of imported cases before the day being considered.
Details may be found in the accompanying Technical Report; the code is available here.
[1]: Fraser, C. (2007). Estimating Individual and Household Reproduction Numbers in an Emerging Epidemic. PLOS ONE 2 (8), https://doi.org/10.1371/journal.pone.0000758.
[2]: Read, J.M., Bridgen, J.R.E., Cummings, D.A.T., Ho, A., Jewell, C.P. (2020) Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions. MedRxiv, Version 2, 01/28/2020, https://doi.org/10.1101/2020.01.23.20018549.